Finite Group Databasebeta

← #20285098

Subgroups — #20285098

Order 6784.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811179C211yesmaximal, characteristic, complemented
2#124×D53C211yesmaximal, characteristic
3#16810984C212yesmaximal, complemented
4#16810981C212yesmaximal, complemented
5#16810980C211yesmaximal, characteristic, complemented
6#16810978C211yesmaximal, characteristic, complemented
7#16810977C211yesmaximal, characteristic, complemented
8#16810630C212yesmaximal, complemented
9#16810626C212yesmaximal
10C4×#11719912C211yesmaximal, characteristic, complemented
11C53×#488C211yesmaximal, characteristic, complemented
12C2×Q8×D53C2211yescharacteristic
13C2×D4×D53C2211yescharacteristic
14C2×#11719912C2212yes—
15#13133216C2212yescomplemented
16#13133209C2211yescharacteristic, complemented
17#13133206C2211yescharacteristic
18#13133205C2211yescharacteristic
19#13133204C2211yescharacteristic
20#13133203—22no—
21#13133195—22no—
22#13133182—22no—
23#13133175C2212yes—
24C22⋊C4×D53—22no—
25C42×D53C2211yescharacteristic
26#13133142C2212yescomplemented
27#13133141C2212yescomplemented
28#13133140C2212yescomplemented
29#13133139C2212yes—
30#13133138C2212yes—
31#13133137C2212yes—
32#13133127C2212yescomplemented
33C4×#88322C2211yescharacteristic, complemented
34#125×C53C2211yescharacteristic, complemented
35#124×C53C2211yescharacteristic
36#103×C53C2212yescomplemented
37#102×C53C2212yes—
38C4×C8×C53C2211yescharacteristic, complemented
39C23×D53—21no—
40C2×Q8×C53C2311yescharacteristic
41C2×D4×C53C2311yescharacteristic
42C2×C22⋊D53—21no—
43Q8×D53—24no—
44D4×D53—24no—
45C2×D212C2311yescharacteristic
46C2×C4×D53—21no—
47C2×C4×D53D412yes—
48C2×C4×D53C2311yescharacteristic
49C2×Dic106—22no—
50C2×Dic106—21no—
51C2×Dic106C2311yescharacteristic
52C2×C8×C53C2312yes—
53C4⋊C4×C53—22no—
54C4⋊C4×C53C2312yes—
55C22⋊C4×C53—22no—
56C42×C53C2311yescharacteristic
57C22⋊Dic53—22no—
58#11719921—22no—
59#11719921—22no—
60#11719921—22no—
61C4⋊Dic53C2312yes—
62#11719919—22no—
63#11719919—22no—
64C4×Dic53—22no—
65C4×Dic53D412yes—
66C4×Dic53C2311yescharacteristic
67C2×#88322C2312yes—
68#11719912—24no—
69C23×C53—21no—
70C22×D53—21no—
71C22×D53—44no—
72C22×D53—42no—
73C22×D53C2×D411yescharacteristic
74Q8×C53—22no—
75D4×C53—22no—
76C2×C4×C53—21no—
77C2×C4×C53—22no—
78C2×C4×C53C2×D412yes—
79C2×C4×C53C2411yescharacteristic
80C22⋊D53—42no—
81C2×Dic53—21no—
82C2×Dic53—22no—
83C2×Dic53C2×D412yes—
84C2×Dic53C2×D411yescharacteristic
85D212—22no—
86C4×D53—42no—
87C4×D53—42no—
88C4×D53#4114yes—
89Dic106—22no—
90Dic106—44no—
91Dic106—42no—
92C8×C53—22no—
93#88322—22no—
94C22×C53—41no—
95C22×C53—42no—
96C22×C53C22×D411yescharacteristic
97C2×D53—22no—
98C2×D53—41no—
99C2×D53—42no—
100C2×D53—84no—
101C2×D53#12412yes—
102C4×C53—22no—
103C4×C53—41no—
104C4×C53—42no—
105C4×C53C2×#4112yes—
106Dic53—22no—
107Dic53—41no—
108Dic53—42no—
109Dic53#12412yes—
110#3140—531nomaximal, Sylow
111C2×C53—81no—
112C2×C53C2×#12411yescharacteristic
113C2×C53#57512yes—
114D53—22no—
115D53—81no—
116#533—531no—
117C2×#124—531no—
118#489—532no—
119#489—532no—
120#488—531no—
121#488—531no—
122#488—531no—
123#488D5311yescharacteristic, complemented
124#419—532no—
125#418—532no—
126C4×M16—531no—
127C53#314011yescharacteristic, complemented, Sylow
128C22×Q8—531no—
129C22×D4—531no—
130C2×M16—532no—
131C4⋊Q8—532no—
132#125—531no—
133#125C2×D5311yescharacteristic, complemented
134#124—531no—
135#124—531no—
136#124—531no—
137#124—1062no—
138#124C2×D5311yescharacteristic
139C22⋊Q8—1062no—
140C4×Q8—1062no—
141#117—532no—
142C2×C22⋊C4—1062no—
143C2×C42—531no—
144#103—532no—
145#103—532no—
146#103—532no—
147#103C2×D5312yescomplemented
148#102—532no—
149#102—532no—
150#102—532no—
151#102C2×D5312yes—
152#97—532no—
153C4×C8—531no—
154C4×C8C2×D5311yescharacteristic, complemented
155C24—1061no—
156C2×Q8—531no—
157C2×Q8—1062no—
158C2×Q8—1061no—
159C2×Q8—1064no—
160C2×Q8C22×D5311yescharacteristic
161C2×D4—531no—
162C2×D4—1064no—
163C2×D4—1061no—
164C2×D4C22×D5311yescharacteristic
165C22×C4—531no—
166C22×C4—532no—
167C22×C4—1061no—
168M16—1064no—
169C2×C8—532no—
170C2×C8C22×D5312yes—
171C4⋊C4—22no—
172C4⋊C4—532no—
173C4⋊C4—1062no—
174C4⋊C4—1062no—
175C4⋊C4C22×D5312yes—
176C22⋊C4—22no—
177C22⋊C4—1062no—
178C22⋊C4—1062no—
179C22⋊C4—1062no—
180C22⋊C4—1062no—
181C42—532no—
182C42—531no—
183C42—1062no—
184C42C22×D5311yescharacteristic
185C23—21no—
186C23—531no—
187C23—1061no—
188C23—2122no—
189C23—2124no—
190Q8—22no—
191Q8—1062no—
192Q8—2122no—
193Q8—2124no—
194D4—22no—
195D4—1062no—
196D4—2122no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—531no—
200C2×C4—534no—
201C2×C4—532no—
202C2×C4—1062no—
203C2×C4—1061no—
204C2×C4—2122no—
205C2×C4—2122no—
206C2×C4D4×D5312yes—
207C2×C4C23×D5311yescharacteristic
208C8—22no—
209C8—1062no—
210C22—41no—
211C22—42no—
212C22—532no—
213C22—1062no—
214C22—4244no—
215C22—2121no—
216C22—2122no—
217C22C2×D4×D5311yescharacteristic
218C4—22no—
219C4—41no—
220C4—42no—
221C4—532no—
222C4—1062no—
223C4—2121no—
224C4—2122no—
225C4#41×D5312yes—
226C2—81no—
227C2—1062no—
228C2—4241no—
229C2#124×D5311yescharacteristic
230C2#1681142012yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#488C2
socleC22×C53C22×D4
hypercenter#488D53
hypocenterC53#3140
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #16811179C2semidirect
3. #16810984C2semidirect
4. #16810981C2semidirect
5. #16810980C2semidirect
6. #16810978C2semidirect
7. #16810977C2semidirect
8. #16810630C2semidirect
10. C4×#11719912C2semidirect
11. C53×#488C2semidirect
15. #13133216C22semidirect
16. #13133209C22semidirect
26. #13133142C22semidirect
27. #13133141C22semidirect
28. #13133140C22semidirect
32. #13133127C22semidirect
33. C4×#88322C22semidirect
34. #125×C53C22semidirect
36. #103×C53C22semidirect
38. C4×C8×C53C22semidirect
123. #488D53semidirect
127. C53#3140semidirect
133. #125C2×D53semidirect
147. #103C2×D53semidirect
154. C4×C8C2×D53semidirect